| L(s) = 1 | − 0.176·2-s − 1.96·4-s − 5-s + 7-s + 0.701·8-s + 0.176·10-s − 2.87·11-s − 0.00387·13-s − 0.176·14-s + 3.81·16-s − 17-s + 6.65·19-s + 1.96·20-s + 0.508·22-s − 6.71·23-s + 25-s + 0.000683·26-s − 1.96·28-s + 3.59·29-s + 2.71·31-s − 2.07·32-s + 0.176·34-s − 35-s − 8.02·37-s − 1.17·38-s − 0.701·40-s + 6.98·41-s + ⋯ |
| L(s) = 1 | − 0.124·2-s − 0.984·4-s − 0.447·5-s + 0.377·7-s + 0.247·8-s + 0.0558·10-s − 0.867·11-s − 0.00107·13-s − 0.0472·14-s + 0.953·16-s − 0.242·17-s + 1.52·19-s + 0.440·20-s + 0.108·22-s − 1.39·23-s + 0.200·25-s + 0.000134·26-s − 0.372·28-s + 0.668·29-s + 0.488·31-s − 0.367·32-s + 0.0303·34-s − 0.169·35-s − 1.31·37-s − 0.190·38-s − 0.110·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9648680935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9648680935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.176T + 2T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.00387T + 13T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 0.0502T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 - 6.02T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.037043814546432680885715583868, −7.83182620574880347583404398940, −6.87865617483158800556650267618, −5.86161694280861954764096934501, −5.10436642351862280275823089117, −4.64894492385625627459544222165, −3.71834548161267880205792795983, −3.00900204269724467409755239158, −1.74191956607680071562885822546, −0.54819858854146726145005955365,
0.54819858854146726145005955365, 1.74191956607680071562885822546, 3.00900204269724467409755239158, 3.71834548161267880205792795983, 4.64894492385625627459544222165, 5.10436642351862280275823089117, 5.86161694280861954764096934501, 6.87865617483158800556650267618, 7.83182620574880347583404398940, 8.037043814546432680885715583868
